Vector-valued fractal functions: Fractal dimension and Fractional calculus

TL;DR

This paper explores the dimensional properties of vector-valued fractal interpolation functions and their fractional integrals, revealing differences from real-valued cases and providing bounds for Hausdorff dimensions.

Contribution

It introduces new bounds and results for the Hausdorff dimension of vector-valued fractal functions and their fractional integrals, highlighting differences from real-valued functions.

Findings

Bounds for Hausdorff dimension of vector-valued fractal functions' graphs

Bounds for Hausdorff dimension of associated invariant measures

Dimensional results for fractional integrals of vector-valued fractal functions

Abstract

There are many research available on the study of real-valued fractal interpolation function and fractal dimension of its graph. In this paper, our main focus is to study the dimensional results for vector-valued fractal interpolation function and its Riemann-Liouville fractional integral. Here, we give some results which ensure that dimensional results for vector-valued functions are quite different from real-valued functions. We determine interesting bounds for the Hausdorff dimension of the graph of vector-valued fractal interpolation function. We also obtain bounds for the Hausdorff dimension of associated invariant measure supported on the graph of vector-valued fractal interpolation function. Next, We discuss more efficient upper bound for the Hausdorff dimension of measure in terms of probability vector and contraction ratios. Furthermore, we determine some dimensional results for the graph of the Riemann-Liouville fractional integral of a vector-valued fractal interpolation function.